Theorem riotasv2s 39255

Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5349) in the form of a substitution instance. Special case of riota2f 7341. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypothesis

Ref	Expression
riotasv2s.2	⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))

Assertion

Ref	Expression
riotasv2s	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐸,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2s

Step	Hyp	Ref	Expression
1		3simpc 1151	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → (𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)))
2		simp1 1137	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐴 ∈ 𝑉)
3		riotasv2s.2	. . . . . 6 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
4		nfra1 3261	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
5		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
6	4, 5	nfriota 7329	. . . . . 6 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
7	3, 6	nfcxfr 2897	. . . . 5 ⊢ Ⅎ𝑦𝐷
8	7	nfel1 2916	. . . 4 ⊢ Ⅎ𝑦 𝐷 ∈ 𝐴
9		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦 𝐸 ∈ 𝐵
10		nfsbc1v 3761	. . . . 5 ⊢ Ⅎ𝑦[𝐸 / 𝑦]𝜑
11	9, 10	nfan 1901	. . . 4 ⊢ Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)
12	8, 11	nfan 1901	. . 3 ⊢ Ⅎ𝑦(𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑))
13		nfcsb1v 3874	. . . 4 ⊢ Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶
14	13	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶)
15	10	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑)
16	3	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
17		sbceq1a 3752	. . . 4 ⊢ (𝑦 = 𝐸 → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
18	17	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
19		csbeq1a 3864	. . . 4 ⊢ (𝑦 = 𝐸 → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
20	19	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
21		simpl 482	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 ∈ 𝐴)
22		simprl 771	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐸 ∈ 𝐵)
23		simprr 773	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑)
24	12, 14, 15, 16, 18, 20, 21, 22, 23	riotasv2d 39254	. 2 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝐴 ∈ 𝑉) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)
25	1, 2, 24	syl2anc 585	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052  [wsbc 3741  ⦋csb 3850  ℩crio 7316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-riotaBAD 39250

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-riota 7317  df-undef 8217

This theorem is referenced by: (None)